Midwest cousins of Barnes-Wall lattices

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks $2^{d-1}\pm 2^{d-k-1}$, for odd integers $d\ge 3$ and integers $k=1,2,\mathrm{.}\mathrm{.}\mathrm{.},\frac{d-1}{2}$. Their minimum norms are moderately high: $2^{\lfloor \frac{d}{2}\rfloor -1}$.